International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 129126, 14 pages
Research Article

Old and New Identities for Bernoulli Polynomials via Fourier Series

1Departamento de Matemáticas, Universidad de Salamanca, 37008 Salamanca, Spain
2Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
3Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain

Received 20 March 2012; Accepted 4 May 2012

Academic Editor: Ricardo Estrada

Copyright © 2012 Luis M. Navas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Bernoulli polynomials 𝐵 𝑘 restricted to [ 0 , 1 ) and extended by periodicity have nth sine and cosine Fourier coefficients of the form 𝐶 𝑘 / 𝑛 𝑘 . In general, the Fourier coefficients of any polynomial restricted to [ 0 , 1 ) are linear combinations of terms of the form 1 / 𝑛 𝑘 . If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.